CRB-Secrecy Rate Region in ISAC

• CRB-secrecy region is a composite metric that jointly constrains CRB-based sensing performance and secrecy rates in systems like MIMO ISAC.
• It leverages advanced transmit covariance and hybrid beamforming optimizations to manage the nonconvex trade-off between sensing accuracy and confidentiality.
• Research insights indicate that achieving Pareto-optimal designs requires full-rank covariance matrices and iterative techniques such as WMMSE reformulation.

The expression CRB-Secrecy Rate Region is not a standard title in the cited literature. The underlying theory instead appears in two adjacent strands: CRB-rate regions for integrated sensing and communication, where sensing quality is quantified through a Cramér–Rao bound (CRB), and secrecy-rate or secrecy-capacity regions for broadcast, interference, and multiple-access channels, where confidentiality is quantified through equivocation or strong-secrecy leakage. This suggests an Editor's term, “CRB-secrecy region”, for a feasible or Pareto-optimal set that would jointly constrain a CRB-based sensing metric and a secrecy-rate metric under transmit covariance, power, or beamforming constraints (Hua et al., 2022, Zhou et al., 17 Feb 2025), [0702099].

1. Terminological status and formal template

The most explicit bridge appears in the MIMO ISAC literature. In the point-to-point MIMO setting, the CRB-rate region is defined as the set of all simultaneously achievable sensing-communication pairs under a power constraint, with the Pareto boundary obtained by maximizing rate subject to a CRB ceiling (Hua et al., 2022). In the near-field RSMA-enabled ISAC setting, the analogous performance region pairs the max-min user rate with the reciprocal sensing metric $$1/\operatorname{Tr}(\mathrm{CRB}(\mathbf r,\boldsymbol\theta))$$, and the Pareto boundary is traced by solving a sensing-centric minimization under communication-rate constraints (Zhou et al., 17 Feb 2025).

The same MIMO ISAC analysis also states that one can define a CRB-secrecy region by replacing ordinary rate with secrecy rate: $$\max_{\mathbf{Q}} \; R_{\text{sec}}(\mathbf{Q}) \quad \text{s.t.} \quad \mathrm{CRB}(\mathbf{Q})\le \Gamma,\; \operatorname{tr}(\mathbf{Q})\le P.$$ The paper immediately notes that the optimization becomes harder because secrecy rate typically involves a difference of two log-determinants, so a single SVD does not in general diagonalize both legitimate and eavesdropper channels (Hua et al., 2022). This establishes the concept as a natural extension, but not yet as a fully standardized theory.

Literature strand	Representative region	Role in the composite notion
CRB-rate ISAC	$\mathcal{C}_{C-R}(P)$, or achievable pairs $(\hat R,\hat\Phi)$ (Hua et al., 2022, Zhou et al., 17 Feb 2025)	Supplies the CRB-based sensing side
Secrecy-rate information theory	Secrecy capacity or achievable secrecy regions for BC, IC, MAC, and related models [0702099], (0806.4200, 0910.3658)	Supplies the confidentiality side
Possible synthesis	$\max_{\mathbf Q} R_{\text{sec}}(\mathbf Q)$ under a CRB constraint (Hua et al., 2022)	Suggests a CRB-secrecy Pareto formulation

2. CRB-based achievable regions in ISAC

In the point-to-point MIMO ISAC formulation, the base station uses a unified transmit covariance matrix

$$\mathbf{Q}=\mathbb{E}\{\mathbf{x}(n)\mathbf{x}^H(n)\}\succeq \mathbf{0},$$

with power budget $\operatorname{tr}(\mathbf Q)\le P$. The communication rate is

$$R(\mathbf{Q})=\log_2\det\!\left(\mathbf{I}_{N_c}+\frac{1}{\sigma_c^2}\mathbf{H}_c\mathbf{Q}\mathbf{H}_c^H\right),$$

while the sensing metric is the trace of the CRB matrix,

$$\mathrm{CRB}(\mathbf{Q})=\frac{\sigma_s^2N_s}{L}\operatorname{tr}(\mathbf{Q}^{-1}).$$

The resulting CRB-rate region is

$$\mathcal{C}_{C-R}(P) \triangleq \left\{(\bar{\Gamma},\bar{R}): \bar{\Gamma}\ge \frac{\sigma_s^2N_s}{L}\operatorname{tr}(\mathbf{Q}^{-1}), \; \bar{R}\le \log_2\det\!\left(\mathbf{I}_{N_c}+\frac{1}{\sigma_c^2}\mathbf{H}_c\mathbf{Q}\mathbf{H}_c^H\right), \; \operatorname{tr}(\mathbf{Q})\le P,\; \mathbf{Q}\succeq \mathbf{0} \right\}.$$

A central structural result is that the optimal transmit covariance on the Pareto boundary is full rank, because finite CRB requires $$\max_{\mathbf{Q}} \; R_{\text{sec}}(\mathbf{Q}) \quad \text{s.t.} \quad \mathrm{CRB}(\mathbf{Q})\le \Gamma,\; \operatorname{tr}(\mathbf{Q})\le P.$$0 (Hua et al., 2022).

The near-field RSMA-enabled formulation generalizes this region to multiuser communication fairness and multi-target sensing. The achievable region pairs

$$\max_{\mathbf{Q}} \; R_{\text{sec}}(\mathbf{Q}) \quad \text{s.t.} \quad \mathrm{CRB}(\mathbf{Q})\le \Gamma,\; \operatorname{tr}(\mathbf{Q})\le P.$$1

subject to power, common-rate, and hybrid analog-digital beamforming constraints. The use of the reciprocal CRB is explicit: it keeps the sensing metric increasing with better sensing performance and places the region in the first quadrant (Zhou et al., 17 Feb 2025).

The sensing model in that work is intrinsically near-field. The array response depends on both distance and angle, and the CRB is derived for the stacked parameter vector

$$\max_{\mathbf{Q}} \; R_{\text{sec}}(\mathbf{Q}) \quad \text{s.t.} \quad \mathrm{CRB}(\mathbf{Q})\le \Gamma,\; \operatorname{tr}(\mathbf{Q})\le P.$$2

with

$$\max_{\mathbf{Q}} \; R_{\text{sec}}(\mathbf{Q}) \quad \text{s.t.} \quad \mathrm{CRB}(\mathbf{Q})\le \Gamma,\; \operatorname{tr}(\mathbf{Q})\le P.$$3

This formulation is important because it identifies the CRB side of any future CRB-secrecy construction not merely as an abstract sensing metric, but as a Fisher-information-derived bound tied to the transmit covariance $$\max_{\mathbf{Q}} \; R_{\text{sec}}(\mathbf{Q}) \quad \text{s.t.} \quad \mathrm{CRB}(\mathbf{Q})\le \Gamma,\; \operatorname{tr}(\mathbf{Q})\le P.$$4 and to derivatives of near-field steering vectors with respect to distance and angle (Zhou et al., 17 Feb 2025).

3. Secrecy-rate regions and secrecy-capacity regions

The secrecy side of the composite notion is much older and more developed. In discrete memoryless interference and broadcast channels with independent confidential messages, secrecy is measured by the equivocation rate at the eavesdropping receiver, and inner and outer bounds on secrecy capacity regions are derived. The outer bounds have an identical mutual information expression for both interference and broadcast models, with the distinction arising from the input distributions over which the expression is optimized. The inner bounds are achieved by random binning, and for the broadcast channel a double-binning coding scheme allows both joint encoding and preserving of confidentiality. For the special switch channel, the bounds meet. For Gaussian interference channels, several achievable schemes are described, including an encoding scheme in which transmitters dedicate some of their power to create artificial noise, and this scheme is shown to outperform both time-sharing and simple multiplexed transmission of the confidential messages [0702099].

In the broadcast channel with an external eavesdropper, secrecy is again expressed through equivocation: $$\max_{\mathbf{Q}} \; R_{\text{sec}}(\mathbf{Q}) \quad \text{s.t.} \quad \mathrm{CRB}(\mathbf{Q})\le \Gamma,\; \operatorname{tr}(\mathbf{Q})\le P.$$5 For the general non-degraded case, an inner bound is established using random binning together with Gelfand–Pinsker binning, matching Marton’s inner bound when confidentiality is removed. For the degraded case, the secrecy capacity region is characterized exactly, with the Secret Superposition Scheme based on Cover’s superposition coding and random binning. In the AWGN degraded case, the secret superposition scheme with Gaussian codebook is optimal, and the converse relies on the generalized entropy power inequality (0806.4200).

A closely related formulation for the broadcast channel with an eavesdropper introduces common and private layers through auxiliaries $$\max_{\mathbf{Q}} \; R_{\text{sec}}(\mathbf{Q}) \quad \text{s.t.} \quad \mathrm{CRB}(\mathbf{Q})\le \Gamma,\; \operatorname{tr}(\mathbf{Q})\le P.$$6, $$\max_{\mathbf{Q}} \; R_{\text{sec}}(\mathbf{Q}) \quad \text{s.t.} \quad \mathrm{CRB}(\mathbf{Q})\le \Gamma,\; \operatorname{tr}(\mathbf{Q})\le P.$$7, and $$\max_{\mathbf{Q}} \; R_{\text{sec}}(\mathbf{Q}) \quad \text{s.t.} \quad \mathrm{CRB}(\mathbf{Q})\le \Gamma,\; \operatorname{tr}(\mathbf{Q})\le P.$$8, and derives both an inner bound for the general non-degraded case and the exact secrecy capacity region for the degraded case. In the degraded setting,

$$\max_{\mathbf{Q}} \; R_{\text{sec}}(\mathbf{Q}) \quad \text{s.t.} \quad \mathrm{CRB}(\mathbf{Q})\le \Gamma,\; \operatorname{tr}(\mathbf{Q})\le P.$$9

for some distribution $\mathcal{C}_{C-R}(P)$0. The Gaussian specialization yields explicit difference-of-capacities formulas and shows that Secret Superposition Scheme with Gaussian codebook is optimal (0910.3658).

Other secrecy-region models broaden the range of mechanisms and constraints. In SWIPT wiretap interference channels, the secrecy rate region is characterized under receiver energy harvesting constraints, and the Pareto boundary is obtained through a weighted max-min optimization over transmit powers and power-splitting coefficients; the resulting problem is a signomial program, which is relaxed iteratively into a geometric program (Kariminezhad et al., 2017). In degraded broadcast channels with receiver message side information and non-causal transmitter CSI, inner and outer bounds establish secrecy capacity regions for both complementary and non-complementary RMSI, and the Gaussian specialization shows that the combination of CSI and RMSI can enlarge secrecy rates (Pakravan et al., 24 Jun 2026).

Not all secrecy regions are equivocation-based. In the cribbing MAC, secrecy is formulated under strong secrecy constraints using

$\mathcal{C}_{C-R}(P)$1

and channel resolvability is used to derive achievable secrecy rate regions. For degraded message sets, non-causal cribbing, and causal cribbing, the corresponding resolvability regions are exactly characterized; for strictly-causal cribbing, inner and outer bounds are provided, with the achievability based on block-Markov coding (Helal et al., 2018).

4. Boundary-achieving mechanisms and optimization methods

The coding and optimization techniques on the two sides of the proposed composite notion are structurally different, but they are united by boundary characterization under coupled constraints.

On the secrecy side, the foundational tools are random binning, double-binning, Gelfand–Pinsker binning, Marton coding, superposition coding, and Secret Superposition Scheme. In multiuser secrecy problems these mechanisms create randomness, correlate layers, and suppress leakage terms of the form $\mathcal{C}_{C-R}(P)$2 or conditional variants thereof [0702099], (0806.4200, 0910.3658). In cooperative or feedback-like settings, secrecy can also be generated through compress-and-forward, joint jamming and relaying, or cooperative resolvability, the latter linking small KL divergence at the eavesdropper to vanishing strong-secrecy leakage (0811.1317, Helal et al., 2018).

On the CRB side, the dominant mechanisms are covariance or beamformer optimization. For point-to-point MIMO ISAC, the Pareto boundary is characterized via a CRB-constrained rate maximization. After SVD of the communication channel, the optimal covariance is diagonal in the singular-vector basis and reduces to explicit power allocation over communication subchannels and sensing-only null-space subchannels; the sensing-only powers are equal, and the communication-mode powers follow a KKT-derived cubic-root expression (Hua et al., 2022). For near-field RSMA-enabled ISAC, the boundary is traced by a constrained sensing minimization under user-rate thresholds, with the coupled hybrid beamforming variables handled by a PDD-based double-loop algorithm, block coordinate descent, and a WMMSE reformulation. A lower-complexity two-stage design fixes the analog beamformer heuristically and optimizes the digital beamformer afterward (Zhou et al., 17 Feb 2025).

The secrecy literature also contains its own nonconvex boundary-tracing procedures. In SWIPT wiretap interference channels, the Pareto boundary is formulated as a weighted max-min problem. The nonconvex secrecy and energy-harvesting constraints are written as signomial inequalities, and single condensation based on the arithmetic-geometric mean inequality converts the problem into a geometric program solved iteratively (Kariminezhad et al., 2017). This is methodologically close to the CRB-rate literature’s constrained-boundary viewpoint, even though the underlying metrics are different.

A plausible implication is that a genuine CRB-secrecy region would inherit both classes of techniques: secrecy coding or secrecy-aware signaling to shape the eavesdropper’s observation, and covariance or beamformer optimization to keep the CRB within an admissible range.

5. Structural insights, special cases, and common misconceptions

A first structural insight is that the CRB side imposes a full-rank requirement that has no direct analogue in classical secrecy-capacity formulas. In MIMO ISAC, if the rate-maximizing covariance $\mathcal{C}_{C-R}(P)$3 is rank deficient, then $\mathcal{C}_{C-R}(P)$4, which means that a communication-optimal design may be unusable for sensing (Hua et al., 2022). This directly warns against the misconception that a secrecy-optimal or rate-optimal covariance is automatically sensing-feasible.

A second insight is that secrecy and cooperation do not stand in simple opposition. In cooperative relay broadcast channels, user cooperation can increase the achievable secrecy region. The achievable scheme combines Marton coding and compress-and-forward, and in Gaussian channels both users can have positive secrecy rates, which is not possible for scalar Gaussian broadcast channels without cooperation (0811.1317). Likewise, in interference settings, secrecy can be improved by deliberately dedicating power to artificial noise [0702099].

A third insight concerns the geometry of achievable regions. In SWIPT wiretap interference channels, the secrecy region can become non-convex, especially as interference grows, and time-sharing may be needed to achieve the full boundary (Kariminezhad et al., 2017). In slotted multiple-access wiretap channels, the long-term secrecy-rate region can instead be enlarged to the ordinary MAC Shannon capacity region by using previous confidential messages as keys and maintaining secret-key buffers; the same approach extends to ergodic fading MAC-WT (Shah et al., 2015). These examples show that region shape depends as much on protocol architecture and resource reuse as on the underlying channel law.

A fourth insight is that the CRB formulation itself is not unique. One line of work uses the trace of the CRB matrix as the sensing metric (Hua et al., 2022), while another uses its reciprocal trace so that better sensing corresponds to a larger coordinate in the achievable region (Zhou et al., 17 Feb 2025). This matters for any CRB-secrecy formulation because the ordering of Pareto-optimal points depends on which scalarization of the CRB is adopted.

A final misconception is to treat existing CRB-rate and secrecy-rate results as already constituting a unified CRB-secrecy theory. The cited literature does not yet provide such a complete characterization. What exists are mature secrecy-region results, mature CRB-rate results, and an explicit suggestion that the latter can be extended by replacing ordinary rate with secrecy rate (Hua et al., 2022).

6. Research status and likely synthesis directions

The present state of the literature is asymmetric. Secrecy-rate regions are well developed for discrete memoryless broadcast, interference, relay-broadcast, cribbing multiple-access, SWIPT interference, and degraded broadcast channels with CSI or receiver message side information [0702099], (0811.1317, Helal et al., 2018, Pakravan et al., 24 Jun 2026). CRB-rate regions are well developed for point-to-point MIMO ISAC with extended targets and for RSMA-enabled near-field ISAC with hybrid analog-digital beamforming (Hua et al., 2022, Zhou et al., 17 Feb 2025). What is missing is a paper in the cited set that fully characterizes a CRB-secrecy rate region in the strict sense.

The MIMO ISAC framework nevertheless provides a concrete starting point. It already states that one can define a CRB-secrecy region by optimizing secrecy rate subject to CRB and power constraints, and it also identifies the main technical obstruction: the secrecy objective is a difference of two log-determinants, so the clean SVD decoupling available for ordinary rate is generally lost (Hua et al., 2022). The near-field RSMA literature adds that hybrid beamforming, common-stream design, and near-field spherical-wave modeling can materially reshape the CRB-rate Pareto boundary (Zhou et al., 17 Feb 2025). The secrecy literature adds that randomization, superposition, artificial noise, cooperative relaying, resolvability, and side information can materially reshape secrecy regions [0702099], (0811.1317, Helal et al., 2018, Pakravan et al., 24 Jun 2026).

This suggests that a future CRB-secrecy theory would likely be a Pareto-boundary problem with full-rank sensing constraints and non-separable secrecy objectives, solved by some mixture of semidefinite optimization, generalized eigenmode design, WMMSE-type reformulation, or iterative penalty methods, together with secrecy-aware signaling or coding. That synthesis is strongly motivated by the cited results, but it remains, in the current corpus, an extension rather than a completed classification theorem (Hua et al., 2022).